I am familiar with the trail at Picacho Peak, although my info may be dated.  I never needed gloves.  I. am sure many people never grab the cables at all.  The trail is wide and comfortable.  The backside, the side not facing the interstate, is significantly more gentle.

I would recommend any light work glove, synthetic fabric with light leather patches, such as you can find at any big box hardware store (or even little box stores).  Stout leather gloves would work well and would be useful around cactus.

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